6. Normal Distribution & Continuous Random Variables

Uniform Distribution

6. Normal Distribution & Continuous Random Variables

Uniform Distribution

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Multiple Choice

Shade the area corresponding to the probability listed, then find the probability.
$P\left(2<X<4\right)$

Graph showing a uniform distribution with shaded area between x=2 and x=4, indicating the probability density. ; $P\left(2<X<4\right)=0.25$

Graph showing a uniform distribution with a shaded area between x=2 and x=4, indicating the probability density. ; $P\left(2<X<4\right)=0.25$

Graph showing a uniform distribution with shaded area between x=2 and x=4, indicating the probability density. ; $P\left(2<X<4\right)=0.5$

Graph showing a uniform distribution with a shaded area between x=2 and x=4, indicating the probability density. ; $P\left(2<X<4\right)=0.5$

Verified step by step guidance

Step 1: Identify the type of probability distribution represented in the graph. The graph shows a uniform probability density function, where the probability density is constant (0.25) across the range of x values from 1 to 5.

Step 2: Understand the problem. The goal is to calculate the probability for the interval P(2 < X < 4). This corresponds to the shaded green area in the graph between x = 2 and x = 4.

Step 3: Recall the formula for calculating probabilities in a uniform distribution. The probability is equal to the area under the curve within the specified interval. For a uniform distribution, the area is calculated as the product of the height of the probability density function and the width of the interval.

Step 4: Determine the width of the interval. The interval is from x = 2 to x = 4, so the width is 4 - 2 = 2.

Step 5: Multiply the width of the interval by the height of the probability density function. The height is given as 0.25, so the probability is calculated as 0.25 × 2. This gives the probability for P(2 < X < 4).